Only one thing you said is right. Why should anything be invertible and why should Ker(A)={0}? If B(v) is in Ker(A) what's A(B(v))? If you think AB=0 then try and put together an argument that will convince me. In your own words.

What's T? I think you MIGHT know what you are trying to say, but that's just a guess. If that's what you are planning to turn in as a solution, I don't think it will work. Can't you put that more clearly?

I don't have your book. What's the linear transformation associated with Bx? Is it also T? What's the linear transformation associated with AB(x). If it's T, I'll scream. You aren't expressing yourself clearly. That's all I'll say.